Page 53 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

40 Preliminaries

1.5 Convex analysis

In this final section we recall the most important results concerning convex
functions.

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Definition 1.49 (i) The set Ω ⊂ R is said to be convex if for every x, y ∈ Ω
and every λ ∈ [0, 1] we have λx +(1 − λ) y ∈ Ω.
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(ii) Let Ω ⊂ R be convex. The function f : Ω → R is said to be convex if
for every x, y ∈ Ω and every λ ∈ [0, 1], the following inequality holds
f (λx +(1 − λ) y) ≤ λf (x)+ (1 − λ) f (y) .

We now give some criteria equivalent to the convexity.
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Theorem 1.50 Let f : R → R and f ∈ C (R ).
(i) The function f is convex if and only if
f (x) ≥ f (y)+ h∇f (y); x − yi , ∀x, y ∈ R n
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where h.; .i denotes the scalar product in R .
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(ii) If f ∈ C (R ),then f is convex if and only if its Hessian, ∇ f,is
positive semi definite.
The following inequality will be important (and will be proved in a particular
case in Exercise 1.5.2).

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Theorem 1.51 (Jensen inequality). Let Ω ⊂ R be open and bounded, u ∈
1
L (Ω) and f : R → R be convex, then
µ Z ¶ Z
1 1
f u (x) dx ≤ f (u (x)) dx .
meas Ω Ω meas Ω Ω
We now need to introduce the notion of duality, also known as Legendre
transform, for convex functions. It will be convenient to accept in the definitions
functions that are allowed to take the value +∞ (a function that takes only finite
values, will be called finite).

Definition 1.52 (Legendre transform). Let f : R n → R (or f : R n →
R ∪ {+∞}).

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(i) The Legendre transform,or dual,of f is the function f : R → R ∪
{+∞} defined by
∗ ∗ ∗
f (x )= sup {hx; x i − f (x)}
x∈R n